Say the Mongol Empire goes through some reformations and stays together, could they end up discovering America.
Well They did build fleets to attack japan, so they can do ship building to some extent. Also they could possibly discover alaska.
They did'nt build fleets, they comandeered the existing Chinese ships (which were mostly Brown water and Green water ships) and then had the Chinese build a bunch of river ships with the intent of using them in the open ocean.
The Mongols were not interested into developing new lands, they were interested in ransacking the old lands. But the again the Bearing straight is only a short skip and a jump across.
This line of reasoning always drives me nuts! Why do people continue to insist that the Chinese were unable to sail blue waters and that the mongols were useless without their horses?
Your description of the invasion fleet is over simplified as well.
The Mongols, like the Romans, took the best of the people they conquered and made use of it.
They had become an accomplished amphibious strike force by the reign of Kublai, as witnessed by the invasions of Japan and Java. They made use of Korean and Vietnamese ships masters, navigators and sailors as well as Chinese.
The Chinese were the most advanced shipwrights of their time (rudder, watertight compartments, the Junk rig etc.) and had been making long voyages for quite some time accumulating great seamanship skills.
The Mongols had moved from nomads on the steppes to conquerors of jungles, deserts and mountains...they were adaptable. and no longer relied on their horses. (the final battle in the conquest of China was a naval battle).
the premise states that the mongol empire survives, that means not just the Yuan Dynasty, but the hordes as well. I believe that that would have made the possibility of trans Pacific discovery less likely, since interest and focus would have been westward and not eastward. But, if we concentrate on the Yuan dynasty surviving and the Mongols becoming more and more sinicized I think it is more possible.
With the maritime skills and technology at their disposal I would not limit the possibility of their discovering the Americas, what they would do with them is another matter
Not the point. The Mongols were looking for conquest and wealth. Why go where ,to the best of knowledge at the time, there is none of that.